Worldwide activities towards the realization of free-electron laser UV and X-ray sources to produce spatially coherent, ultra-short similar to 100 fs pulses with very high peak brilliance (>= 10(28)-10(32) photons/s/ mm(2)/mrad(2)/0.1% BW) are summarized. These sources are based on linear accelerators to overcome the limits to brilliance imposed by the ring geometry. The scientific case includes time-resolved studies of dynamics on sub-ps scales, structural studies by imaging of nonperiodic systems, and investigation of high energy-density phenomena such as nonlinear X-ray optics and the production of warm dense matter. Examples of the existing projects are presented, with emphasis on the presently operational FLASH facility at DESY, which delivers ultrashort coherent pulses at 6.5 nm wavelength. Projects in the US, in Japan, and in Europe, aiming to attain the hard X-ray region, with wavelengths of order 0.1 nm, are described. Plans to control the time and energy structure of the pulses by seeding and harmonic generation schemes are also mentioned.